openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /GraphGeometry /HitEventUpperSet.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.ProductMeasure.FiniteOpenWitness | |
| namespace OAI | |
| section | |
| section | |
| open scoped BigOperators | |
| open scoped Classical ENNReal | |
| open Set | |
| open scoped Classical Topology | |
| open Set MeasureTheory ProbabilityTheory | |
| open scoped Classical | |
| open Filter Topology | |
| open Set MeasureTheory ProbabilityTheory Filter | |
| open scoped Topology Classical | |
| namespace CriticalPercolation.BondGraph | |
| variable {V E : Type*} (G : BondGraph V E) | |
| lemma hitEvent_isUpperSet (o : V) (A : Set V) : IsUpperSet (G.hitEvent o A) := | |
| fun _ _ h => G.hitEvent_mono_edges o A h | |
| lemma hitEvent_hasFiniteOpenWitness (o : V) (A : Set V) : | |
| BernoulliProduct.HasFiniteOpenWitness (G.hitEvent o A) := by | |
| rintro ω ⟨v, hv, hvA⟩ | |
| obtain ⟨s, hs, hsω, hvs⟩ := G.mem_cluster_finite_witness hv | |
| exact ⟨s, hs, hsω, v, hvs, hvA⟩ | |
| lemma lowerSemicontinuous_real_hitEvent [Countable E] (o : V) (A : Set V) : | |
| LowerSemicontinuous (fun p : unitInterval => (G.law p).real (G.hitEvent o A)) := by | |
| simpa [law, BernoulliProduct.law_const] using | |
| BernoulliProduct.lowerSemicontinuous_real_finiteWitness (G.hitEvent o A) | |
| (G.hitEvent_isUpperSet o A) (G.hitEvent_hasFiniteOpenWitness o A) | |
| end CriticalPercolation.BondGraph | |
| open Set MeasureTheory ProbabilityTheory Filter | |
| open scoped Classical Topology | |
| end | |
| end | |
| end OAI | |